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Contradiction as a Form of ContractualIncompleteness�
Dana Hellery and Ran Spieglerz
First version: December 2005This version: September 2006
Abstract
A simple model is presented, in which contradictory instructionsare viewed as a type of contract incompleteness. The model providesa complexity-based rationale for contradictory instructions. If thereare complexity bounds on the contract, there may be an incentive tointroduce contradictions, leaving for another agent the task of inter-preting them. The optimal amount of contradictions depends on thecomplexity bound, the con�ict of interests with the interpreter, andthe institutional constraints on his interpretations. In particular, ahigher complexity bound may result in a larger amount of contradic-tions.
�We are grateful to an editor and a referee for useful comments.yNational Economics Research Associates Inc. E-mail: [email protected] of Economics, Tel Aviv University. E-mail: [email protected]. URL:
http://www.tau.ac.il/~rani.
1
1 Introduction
The contract-theoretic literature has identi�ed two di¤erent notions of con-
tract incompleteness. First, a contract is incomplete if it is not as �ne as it
should be, given veri�ability constraints. Second, a contract is incomplete
if it has lacunae - i.e., it ignores some states of the world. In this paper we
identify a third type of contract incompleteness: contradictory instructions.
To motivate this idea, consider a contract that consists of two clauses: (i) in
case of an earthquake, take action a; (ii) in case of a �re, do not take action a.
This contract has a lacuna, as it fails to specify what happens when neither
of the two events (earthquake, �re) takes place. However, the contract also
contains a contradiction: if both events happen simultaneously, the contract
speci�es mutually contradictory actions.
The contradiction is due to the fact that the contract is a function that
assigns actions to events, rather than mutually exclusive states. Existing
contract-theoretic literature typically formalizes a contract as a function from
states to actions (a couple of exceptions are Battigalli and Maggi (2002) and
Shavell (2006); see below). Such a formalism can capture the familiar types
of contract incompleteness, but it cannot capture contradictions. However,
real-life contracts are written in natural language. A contingency stated
in natural language typically corresponds to an event, rather than to an
individual state. Therefore, it may be more �tting to formalize a contract
as a function from events to actions. Such a formalism can accommodate
contradictions.
In this short paper we study a simple model that provides a rationale
for this third type of contract incompleteness, based on the notion of con-
tracts as functions from events to actions. The rationale is a variant on
the writing-complexity argument for contract incompleteness, due to Dye
(1985), Anderlini and Felli (1994,1999) and Battigalli and Maggi (2002). In
the model, one agent, referred to as the Writer, unilaterally writes down a
list of instructions for another agent, referred to as the Interpreter, to carry
2
out. There is a con�ict of interests between the Writer and the Interpreter.
The state space is [0; 1). An instruction consists of an event - i.e., a subset
of [0; 1) - and an action. Two instructions are contradictory if they specify
intersecting events and di¤erent actions. A lacuna exists if there are states
of the world which are not covered by any instruction.
When a state of nature is covered by exactly one instruction, the In-
terpreter is obliged to take the action that this instruction speci�es in this
state. However, when the state falls into a lacuna or a contradiction, the
Interpreter exercises discretion. We impose a single constraint on the In-
terpreter�s discretion: if two states belong to the same set of events in the
Writer�s instructions list, then the Interpreter must take the same action in
both states.
The following scenario illustrates the model. Suppose that the Writer is
a set of contracting parties or a legislature, whereas the Interpreter is a court
of law. An �instructions list�is a contract or a law. When a state is realized
and gives rise to a lacuna or a contradiction, the court steps in and provides
an interpretation. The constraint we impose on the court�s discretion re�ects
a precedent system. Even if the court can observe the true state, a precedent
system implies that the court must arrive at the same ruling in two states
which are equivalent in terms of the contract or law in question.
If there were no limitations on theWriter�s instructions list, he would want
to write down a complete contract and leave no discretion to the Interpreter,
because of their con�ict of interests. However, we assume that there are two
complexity bounds on the Writer�s instructions list. First, there is an upper
bound K on the number of instructions on the list. Second, the family of
events on which he can draw is restricted. Speci�cally, an admissible event
is a union of no more than r disjoint intervals in [0; 1). Thus, r is a measure
of the richness of the language in which events are phrased.
Complexity measures are inevitably arti�cial, and the present case is no
exception. Nevertheless, we believe that it is reasonable to interpret K and r
3
as measures of the complexity of writing the instructions list. The larger the
number of instructions, and the larger the number of contingencies on which
each instruction conditions, the more costly it is to write the instructions
list.1 As Battigalli and Maggi (2002) state:
�...we have in mind costs that are, broadly speaking, proportional
to the amount of detail in the contract, such as the cost of �guring
out the relevant contingencies and obligations, the cost of think-
ing how to describe them, the cost of time needed to write the
contract, and the cost of lawyers.�(Battigalli and Maggi (2002),
p. 798)
Thus, a key feature of our model is that the Writer�s decision how much
interpretational freedom to assign the court is strategic, and takes writing
costs into account. We believe that this is a realistic feature of the contracting
or legislative process. To quote Shavell (2006):
�We also observe that the courts actively engage in the inter-
pretation of contracts. The courts �ll gaps in contracts, resolve
con�icts and ambiguities of language, and sometimes replace the
parties�express terms with the courts�terms ... Moreover, the
interpretation of contracts is widely understood to in�uence how
parties write contracts: the more closely the courts�interpreted
contracts resemble the parties�true wishes, the more willing the
parties are to leave gaps and to write fairly general terms, whereas
parties are more willing to take extra pains to write more detailed
contracts when courts refrain from interpreting terms or interpret
1Note that we capture writing complexity by placing the upper bounds K and r, ratherthan by specifying a cost function which is increasing in K and r. This is done to facilitateexposition, and does not a¤ect the qualitative results.
4
terms in ways that run counter to their true desires.� (Shavell
(2006), p. 290)
The model studied by Shavell (2006) shares our model�s general structure,
the crucial di¤erence being that it rules out contradictions. Nevertheless, as
the above quote attests, Shavell acknowledges the relevance of contradictions
for the act of interpretation. Indeed, according to Farnsworth (1999), a
substantial proportion of contract trial cases in the American judicial system
are concerned with interpretation, which usually involves overriding terms,
ambiguities of language, and internal contradictions in terms.
Given the complexity constraints, the Writer has an incentive to create
contradictions because they re�ne the e¤ective partition that is induced by
the instructions list. Recall the earthquake-�re example. There are four
states of the world: nothing happens, only an earthquake occurs, only a �re
occurs, both an earthquake and a �re occur. A complete contract specifying
di¤erent actions for di¤erent states would require at least four instructions.
However, if there is no con�ict of interests between the two agents, the two-
clause contract achieves the same outcome. By allowing the Interpreter to
�interpret�contradictions and lacunae, we e¤ectively re�ne the instructions
list. However, when there is a con�ict of interests between the two agents,
the Writer has to trade o¤ this consideration with his desire not to give the
Interpreter too much discretion.
Using a standard Crawford-Sobel model of the con�ict of interests be-
tween the Writer and the Interpreter, we show that the equilibrium degree
of vagueness - i.e., the measure of states that fall into contradictory instruc-
tions or lacunae in subgame perfect equilibrium - is increasing with r. Such a
clear-cut relation does not exist with respect to K. When r = 1, the equilib-
rium degree of vagueness decreases with K. In contrast, when r is very large,
an increase in K may result in a greater degree of vagueness. Nevertheless,
if K is su¢ ciently large, there no lacunae or contradictions in equilibrium.
5
These results may be surprising at �rst glance: one would expect a pri-
ori that a Writer with more �linguistic resources�would leave less for the
Interpreter to interpret, due to their con�ict of interests. The intuition for
our results is that with larger linguistic resources it is possible to generate a
lot more contradictions, thus enabling the Interpreter to re�ne his strategy.
Because the Writer is risk-averse, this strengthens his incentive to introduce
contradictions.
The characterization of the equilibrium instructions list is of particular
interest in the case of r = 1 - i.e., when the events on which instructions
condition are intervals. The K events on the list constitute a pseudo-interval
partition with �fuzzy borders�. This structure has a natural interpretation:
the Writer invokes categories such as �very low�, �moderately low�, �mod-
erately high�, etc., but the exact demarcation of these categories is fuzzy.
The rationale for contradictions relies on the way the Interpreter is al-
lowed to interpret contradictions. We demonstrate this point by examining
two alternative �methods of interpretation�: (i) when a state is covered by
several contradictory instructions, the Interpreter is bound to select one of
the actions speci�ed in these instructions; (ii) there are no constraints at
all on the Interpreter�s discretion in case of lacunae or contradictions. In
both cases, the rationale for contradictions disappears. These observations
demonstrate the importance that norms of contract interpretation have for
contractual incompleteness, and consequently the need for continued explo-
ration of the ways in which courts interpret incomplete contracts.
Related literatureDye (1985) is the �rst paper to model explicitly the writing costs of contracts.
His measure of a contract�s complexity is the number of contingencies on
which the contract conditions (this is analogous to K in the present paper).
However, he assumes that the contract speci�es a partition of the state space,
thereby ruling out both lacunae and contradictions.
Battigalli and Maggi (2002) propose to model contracts as functions of
6
events. They analyze a propositional model of contracts, in which events are
de�ned by disjunctions and conjunctions of elementary propositions and their
negations. However, they exclude contradictions, by focusing on what they
call �feasible�contracts.2 Our paper can be viewed as a modest contribution
to their line of thought.
Shavell (2006) is the closest work to the present paper. Shavell models
a contract as a function of events, yet like Battigalli and Maggi (2002), he
rules out contradiction by assuming that these events are mutually exclusive.
Shavell assumes that contracts are interpreted by the court, and studies the
contracting parties�optimal choice of contracts, given the court�s method of
interpretation. Thus, our paper may be viewed as an extension of Shavell�s
model in the direction of incorporating contradictions into contracts.
Posner (2005) provides a reduced-form, cost-bene�t analysis of contract
interpretation, weighing the negotiation and interpretation costs associated
with a contract. Anderlini, Felli and Postlewaite (2001,2003) are the only
other papers that we are aware of, in which the court plays an active part
in a contract-theoretic model. In their model, the court can void the con-
tract in case of lacunae (which arise from unforeseen contingencies). When a
court voids the contract, the contracting parties can renegotiate. The model
does not accommodate contradictions. These papers study the e¤ects of the
court�s policy on the contracting parties�incentives.
As Lipman (2003) points out, there is a close link between the study of
contradictions in contracts and the study of vagueness in natural language.
Hopefully, further exploration of the former might also shed light on the
latter.2Battigalli-Maggi acknowledge the possibility of contradictions (see footnote 11 in their
paper), and the court�s need to intervene in such cases, but they do not develop this idea.
7
2 The model
Two agents, aWriter and an Interpreter, play a sequential-move game. Their
underlying con�ict of interests is modeled as in the well-known example due
to Crawford and Sobel (1982). Let X = [0; 1) be a set of states of nature.
A subset of X is called an event. The state is drawn from the uniform
probability measure � over [0; 1). The set of actions which are available to
the Interpreter is the set of real numbers R. Player j�s vNM utility function
is given by uj(x; a) = �(x � mj + a)2, where mj 2 R is his �ideal point�.
Without loss of generality, let mwriter = 0 and denote minterpreter = m.
Assume that m 6= 0.The order of moves is as follows. The Writer moves �rst. He chooses an
instructions list, which is a sequence c = (eck; ack)k=1;:::;K , where e
ck is an event
and ack is an action. We refer to ck = (eck; a
ck) as the kth instruction on the
list c. Let Ec = fe1; :::; eKg. The set Ec need not be a partition of X. Weassume that for every k, eck 2 E , where E is a family of admissible events.We introduce E in order to capture bounds on the complexity, or richness,
of the language for describing events. Therefore, we impose some structure
on E . Given that the state space is [0; 1), a natural �primitive�event is aninterval. Thus, we assume that every element in E is a union of no morethan r � 1 disjoint intervals in [0; 1). The larger r, the richer the languagein which events are phrased.3
Before we describe the Interpreter�s move, let us introduce some notation.
For every J � f1; ::; Kg, de�ne
pcJ = fx 2 [0; 1) s.t. x 2 eck 8 k 2 J and x =2 eck 8 k =2 Jg
The set pcJ consists of all states which are covered by the subset of instructions
J in the instructions list c. Note that the collection fpcJgJ�f1;::;Kg is a partition3The intervals may be open or closed. This distinction is immaterial because of the
non-atomicity of �. For notational consistency, we assume that the intervals are half-openof the form [xl; xh).
8
of X: two states belong to the same pcJ if and only if they are covered by
exactly the same subset J of instructions. We refer to the collection of
non-empty sets in fpcJgJ�f1;::;Kg as the e¤ective partition induced by Ec,and denote it by P c. For every x 2 [0; 1), let J c(x) denote the subset ofinstructions that cover x. Formally, J c(x) = fk 2 f1; :::; Kg s.t. x 2 eckg.Whenever jJ c(x)j = 1, x is covered by exactly one instruction - hence,
there is no vagueness regarding the action that the Interpreter needs to take
in state x. When jJ c(x)j = 0, x falls into a lacuna, because it is not coveredby any instruction. When jJ c(x)j > 1, x is covered by multiple instructions.If the actions speci�ed by these instructions are di¤erent, the instructions are
mutually contradictory. Given an instructions list c, let �(c) be the amount
of vagueness that characterizes it. Formally, �(c) is the �-measure of states
x for which jJ c(x)j 6= 1.The Interpreter moves after the Writer has made his choice. Given
(eck; ack)k=1;:::;K , the Interpreter chooses a function I : P
c ! R. That is, theInterpreter�s strategy is measurable with respect to the e¤ective partition
induced by Ec. We impose a single condition on I:
Condition 1 For every state x, if J c(x) 6= ? and ack = a for every k 2J c(x), then I(pcJ) = a.
In particular, if J is a singleton fkg, then I(pcJ) = ack. That is, the
Interpreter�s strategy must coincide with the Writer�s strategy when pcJ does
not represent a lacuna or a contradiction. In other words, the Interpreter is
allowed to exercise discretion only in case of a lacunae or a contradiction.
The model captures situations in which one agent writes down a set of
instructions to be carried out by another agent. The �rst agent can build
contradictions or lacunae into his instructions. On one hand, the contradic-
tions and lacunae give the second agent discretion, which runs against the
�rst agent�s interests. On the other hand, they help the second agent re�ne
9
his strategy, which is favorable to the �rst agent because it reduces decision
errors.
We interpret an instructions list as a law or a contract. The Writer
represents a set of contracting parties, or a legislature. Accordingly, we
may view the Interpreter as a court of law. The court is asked to interpret
contracts and laws when those contain contradictions and lacunae. Under
this interpretation, the assumption that I is measurable with respect to P c
re�ects a �precedent system�. Even if the court knows the state of nature,
the precedent system forces the court to make the same decision in two states
which are indistinguishable in the eyes of the law.
The parameters K and r are indicators of the amount of detail in the
law: K is the number of contingencies on which actions are conditioned, and
r measures the complexity of each contingency. Economists are sometimes
suspicious of the notion of writing costs, as if it involved the mere cost of
ink. However, contracts and laws that contain a greater number of complex
contingencies are more costly to prepare, because the act of putting the
instructions into words is di¢ cult and time consuming.
We conclude this section with a number of comments regarding the in-
terpretation of the model.
Lacunae versus contradictions. The distinction between lacunae and contra-
dictions in our model is quite arti�cial. A lacuna is simply identi�ed as a
particular cell pc? in the e¤ective partition, and treated just like any other
cell pcJ for which jJ j > 1. We could modify our model by allowing the Writerto add a �basket clause�to the instructions list, which assigns a particular
action to any x which belongs to none of the events in the list. The motiva-
tion for such a modi�cation is that the writing costs associated with a basket
clause are negligible, compared to the costs associated with the other instruc-
tions in the list. However, this modi�ed model would be more cumbersome
to analyze than ours, while the qualitative results would be the same.
10
Alternative complexity measures. The basic building blocks in the descrip-
tion of events are intervals. Alternatively, we could assume that a primitive
event is an inequality (x > x0 or x < x0), so that an event in E would be acombination of unions or intersections of inequalities. Since an interval is no
more than an intersection of two inequalities (except for intervals of the form
(0; x0) or (x0; 1), which can be described as a single inequality), our analysis
would be qualitatively the same. Another notion of complexity, employed by
Anderlini and Felli (1994) is based on the concept of computable functions.
The distinction between computable and incomputable contracts is irrele-
vant in the present context. As we shall see below, when K is su¢ ciently
large, the Writer�s incentive to create lacunae and contradictions disappears.
Thus, contract incompleteness in the sense we focus on does not rely on this
distinction, but on the simpler question of the contract�s length.
Further constraints on interpretation. We place no restriction on the Inter-
preter�s discretion in case of lacunae or contradictions, apart from the mea-
surability condition which captures a precedent system. One could imagine
additional constraints. For instance, suppose that when two instructions k
and j specify contradictory actions, ack and acj, for a certain state, the Inter-
preter�s choice of action must re�ect some �averaging� of the two actions.
That is, the Interpreter cannot totally disregard the actions speci�ed by
the con�icting instructions. Such additional constraints on the Interpreter�s
strategy are assumed away, for the sake of expositional simplicity. In Section
4, we discuss alternative �methods of interpretation�.
Costs of interpretation. One could argue that while focusing on writing costs,
we completely ignore the cost of interpreting contradictions and lacunae (see
Posner (2005) for a discussion of such costs). If the Writer does not bear the
interpretation costs, including them will not a¤ect our analysis. Of course,
if the Writer does take into account the costs of interpretation, we need a
more elaborate model to analyze the trade-o¤ between writing costs and
interpretation costs.
11
3 Analysis
In this section we analyze the subgame perfect equilibrium in the game. In
particular, we are interested in the equilibrium degree of vagueness �(c�) -
i.e., the measure of states that fall into lacunae and contradictions - which is
induced by the equilibrium instructions list c� = (e�k; a�k)k=1;:::;K . DenoteE
� =
fe�1; ; ; :; e�Kg, and let P � = fp�JgJ�f1;:::;Kg be the induced e¤ective partition.The players� quadratic utility function has an immediate implication,
which is well-known and therefore not proved here.
Proposition 1 For any set B � [0; 1), let aj(B) � argmaxaEBuj(x + a).
Then, EB[x+ aj(B)] = mj.
That is, if the Interpreter has discretion over some event B, he will choose
the action that adjusts the expected value of x+a to be m. If the Interpreter
lacks discretion over B, the Writer will choose the action that adjusts the
expected value of x+ a to be 0.
This observation greatly facilitates our analysis. In the e¤ective partition
P �, every set p�J for which jJ j 6= 1 is associated with a probability distributionover x + a, whose mean is m. Similarly, every set p�J for which jJ j = 1 is
associated with a probability distribution over x+ a, whose mean is 0. The
Writer faces the following trade-o¤. If he creates contradictions, he re�nes
the e¤ective partition. On one hand, the re�nement reduces the variance of
the probability distribution over x + a. On the other hand, the re�nement
delegates more events to the Interpreter, thereby moving the expected value
of x+ a away from the Writer�s ideal point.
Let us turn to the structure of contradictions and lacunae. The number
of contradictions induced by the equilibrium instructions list is the number
of sets p�J in P� for which jJ j > 1. We will say that E� contains a lacuna if
p�? is non-empty.
12
Proposition 2 Suppose that �(c�) > 0. Then:(i) P � is an intervals partition.
(ii) The number of contradictions induced by c� is the maximal possible, given
K and r.
(iii) E� contains a lacuna.
(iv) All cells p�J with jJ j = 1 have the same measure, and all cells p�J with
jJ j 6= 1 have the same measure.
Proof. Let us prove part (i) �rst. Let c be an instructions list whoseinduced e¤ective partition P c is not an intervals partition. Our �rst step
is to show that c can be transformed into another instructions list, whose
induced e¤ective partition constitutes an intervals partition and contains as
many cells as P c. By assumption, i.e., one of the elements pcJ in Ec is a
union of n > 1 disjoint intervals. Let us try to transform c so that one of
these intervals [x1; x2) is subtracted from pcJ . In order to do so, we need to
select some instruction k 2 J and subtract [x1; x2) from eck. Since n > 1, themodi�ed pcJ remains non-empty.
This modi�cation is infeasible only if it turns eck into a union of more
than r disjoint intervals. But this can occur only if an adjacent interval (say,
[x2; x3), w.l.o.g) belongs to pcL for some L � J . In this case, for every l 2 LnJ ,we can modify ecl by lengthening one of its intervals so that it contains [x1; x2).
Note that this additional modi�cation merely expands one of the intervals
of pcL, without changing the number of disjoint intervals of which it consists.
In this fashion we manage to induce an e¤ective partition which di¤ers from
P c in that the interval [x1; x2) is subtracted from pcJ , without reducing the
number of cells in the e¤ective partition. We can then reiterate this type of
modi�cation, until we get an e¤ective partition which is an intervals partition
and contains as many cells as P c.
To complete the proof of part (i), we need to show that it is optimal
for the Writer to construct an instructions list which induces an e¤ective
partition satisfying two properties: �rst, it is an intervals partition; and
13
second, it contains the largest number of cells that is feasible under (K; r).
By Proposition 1, each of the K cells p�J in P� with jJ j = 1 induces a
probability distribution over x + a, whose mean is 0. Similarly, each cell p�Jin P � with jJ j 6= 1 induces a probability distribution over x+a, whose mean ism. The Writer�s vNM utility function is quadratic. Therefore, conditional on
a delegated event B, and given �(B), the Writer�s expected utility is higher if
B is an interval. Similarly, given that B is an interval, the Writer�s expected
utility conditional on B is higher if �(B) is lower. Therefore, if the largest
e¤ective partition given (K; r) is implementable as an intervals partition, the
Writer will �nd it optimal. Moreover, because the Writer is risk-averse, it is
optimal for him that all intervals p�J with jJ j = 1 have the same measure,
and all intervals p�J with jJ j 6= 1 have the same measure. Obviously, the
largest possible e¤ective partition contains the largest possible number of
contradictions and a lacuna. This also proves parts (ii)-(iv).
The intuition for this result is simple. Suppose that E� is not a partition
- i.e., it delegates a non-empty set of states B to the Interpreter. By Propo-
sition 1, every event that is delegated to the Interpreter induces a probability
distribution over x + a with expected value m. Because the Writer is risk-
averse, it is optimal for him to split B into as many intervals as possible,
thereby reducing the variance of this distribution. This is attained by creat-
ing a lacuna and as many contradictions as possible. The less trivial part of
the proof is to show that it is in fact possible to induce an e¤ective partition
which both contains the largest number of cells that is feasible given (K; r),
and constitutes an intervals partition.
In the remainder of this section, we analyze the cases of r = 1 and r > 1
separately.
14
3.1 The case of r = 1
This special case is of particular interest, for two reasons. First, the family of
admissible events is intuitively the simplest possible, given the state space.
Second, the equilibrium instructions list turns out to have an interesting
structure. We will say that c induces a fuzzy partition if there exists a
numbering of the elements in E, such that: (i) the only allowed intersections
between eck and ecj are as follows: k and j must be consecutive, and neither
interval contains the other; (ii) the lacuna is either [0; x�) or [x�; 1).
In a fuzzy partition, adjacent events are �almost� disjoint, except that
the border between them is blurred. Thus, the e¤ective partition consists of
2K intervals, of which K intervals are delegated to the Interpreter.
Proposition 3 The equilibrium instructions list c� induces a fuzzy partition.All undelegated intervals in P � are of equal measure (1 � �(c�))=K, and alldelegated intervals in P � are of equal measure �(c�)=K.
Proof. By Proposition 2, the e¤ective partition induced by c� is an inter-vals partition. No e¤ective partition can contain more than K undelegated
intervals. A fuzzy partition contains exactly K undelegated intervals, as well
as K delegated intervals. Therefore, all we need to show is that no e¤ec-
tive partition can contain more than 2K intervals. To see why this must
be true, note that since r = 1, every feasible c satis�es eck = [xlk; x
hk] for all
k = 1; :::; K. Therefore, the e¤ective partition is an intervals partition, in
which two adjacent intervals are demarcated by some xlk or xhk. Therefore,
the e¤ective partition cannot contain more than 2K intervals. It follows that
a fuzzy partition is optimal for the Writer. The measure of each delegated
and undelegated cell in the fuzzy partition follows directly from part (iv) of
Proposition 2.
15
Proposition 4 The equilibrium amount of vagueness is
�(c�) = max[1
2� 2K2m2; 0]
Proof. By Proposition 3, the Writer�s expected utility conditional on adelegated interval is �(m2 + �2=12K2), and his expected utility conditional
on an undelegated interval is �(1��)2=12K2. Therefore, �(c�) is the solution
to the following problem:
min�
� � [m2 +�2
12K2] + (1� �) � (1� �)
2
12k2
which yields the desired solution.
The interpretation of this pair of results is attractive, in the sense that
it is suggestive of features discernible in real-life laws and contracts. The
Writer chooses the events in the instructions list so that they constitute a
pseudo-partition with fuzzy borders. It is as if the language that the Writer
employs has a more-or-less clear notion of the meaning of the words �very
bad�, �bad�, �moderately good�, etc., but their demarcation is fuzzy. For
any instruction (e�k; a�k) in the optimal list, the action a
�k is targeted at the
set of �clear-cut�cases, in which the realized state is unquestionably covered
by instruction k only. The ambiguous �borderline� cases are left for the
Interpreter to decide.
The comparative statics are also intuitive. As K decreases (capturing a
situation with high writing costs), and as the con�ict of interests between
the two parties diminishes, the ambiguous borderline cases become more
prevalent. If Km > 12, there is no delegation at all in equilibrium.
3.2 The case of r > 1
What happens to the equilibrium degree of vagueness when we raise r, thus
enriching the Writer�s language?
16
Proposition 5 �(c�) weakly increases with r.
Proof. By Proposition 2, P � is an intervals partition. Moreover, thenumber of undelegated intervals in P � is K. Therefore, when we raise r, we
enable a larger number of delegated intervals without reducing the feasible
number of undelegated intervals. This means that conditional on delega-
tion, the Writer�s expected utility increases, because each delegated interval
becomes narrower. But this means that �(c�) cannot decrease with r.
Thus, as the Writer�s language becomes richer, he chooses to increase the
degree of vagueness. A priori, one could expect that having richer �linguistic
resources�should lead the Writer to delegate less to the Interpreter. How-
ever, a richer language implies that the Writer can create a larger number of
contradictions. More contradictions imply a �ner e¤ective partition, hence
a smaller loss from delegating events to the Interpreter. Note that as long
as �(c�) > 0, the number of contradictions remains the same, in accordance
with Proposition 2.
Let us turn to comparative statics with respect to K. In the previous
sub-section, we saw that when r = 1, �(c�) decreases with K - i.e., that a
longer instructions list implies a lower equilibrium degree of vagueness. It
turns out that this is not a general result. Note that if r is very large, it is
possible to construct an instructions list that induces an e¤ective partition
with 2K cells, such that every cell in the partition corresponds to a di¤erent
element in the power set of K.
Using the same kind of reasoning as in Proposition 3, it can be shown
that P � contains K undelegated intervals of measure (1� �(c�))=K each, as
well as 2K � K delegated intervals of measure �(c�)=(2K � K) each. Theimplication of this feature of P � is that the number of delegated intervals
increases almost exponentially with the length of the instructions list (as long
as r is su¢ ciently large). Thus, an increase in K allows a huge re�nement of
the Interpreter�s strategy, and therefore may raise the Writer�s incentive to
delegate.
17
The following example illustrates how an increase in K can result in an
increase in the equilibrium degree of vagueness. Let m = 115and r = 3, and
compare the cases of K = 2 and K = 3. In both cases, r is su¢ ciently
large so that P � consists of 2K intervals. When K = 2, �(c�) = 0:4644; by
Proposition 2. When K = 3, given a degree of vagueness �, the Writer�s
expected utility conditional on a delegated interval is � 1225� 1
300�2, and his
expected utility conditional on an undelegated interval is � 1108(1 � �)2. It
follows that �(c�) = 0:5.
Although we cannot produce a closed solution for �(c�), we are able to
provide an upper bound. Note that for any (K; r) and any instructions
list c, the Writer�s expected utility conditional on a delegated interval is
bounded from above by �m2. This is a consequence of Proposition 1. The
degree of vagueness that the Writer would choose if this bound could be
attained cannot be lower than the equilibrium degree of vagueness. The
Writer�s expected utility, given � and conditional on an undelegated interval,
is �(1� �)2=12K2. Therefore, the obtain an upper bound on �(c�), we need
to solve the maximization problem
max�
� � (�m2) + (1� �) � �(1� �)2
12K2
which yields
�� = max(1� 2Km; 0) (1)
It follows that max(1�2Km; 0) is an upper bound �(c�) for any (K; r). Thisobservation has the following implication.
Proposition 6 For any r, �(c�) = 0 if Km > 12.
Thus, if K is su¢ ciently large relative to the con�ict of interests, the
Writer will not delegate anything to the Interpreter, regardless of r.
18
3.3 Which type of complexity is more valuable?
Our model relies on two notions of complexity: the number of instructionsK,
and the maximal number r of elementary events on which every instruction
can condition. If one viewed the length of the instructions list (in terms of the
number of words that it contains) as an indicator of its writing complexity,
then Kr would be a reasonable measure. The question naturally arises,
suppose that the Writer were able to manipulate K and r while keeping Kr
�xed, what would he choose to do? Would he prefer a long list of simple
instructions, or a short list of complex instructions? The following result
provides an answer.
Proposition 7 For a �xed Kr, the Writer�s subgame perfect equilibriumpayo¤ attains a maximum when r = 1.
Proof. Fix Kr. Every event e�k 2 E� is a union of r disjoint intervals. Ifwe relabel each of these intervals di¤erently, then it is as if we have K 0 = Kr
and r0 = 1. The e¤ective partition we have constructed is feasible, yet not
necessarily optimal, given (K 0; r0). Therefore, the Writer can only bene�t
when switching from (K; r) to (K 0; r0).
Thus, although both types of complexity are valuable to the Writer, the
number of instructions K is more valuable than the complexity r of every
individual instruction.
4 Alternative constraints on interpretations
The rationale for contradictory instructions in this model is that another
agent will interpret them. Contradictions have an informational content
that enables the two agents to transgress the limits on the instructions list�s
complexity. In this section we explore alternative constraints on the way the
19
Interpreter interprets contradictions, which capture alternative �methods of
interpretation�. We shall see that the rationale for contradictions disappears
under these alternative constraints.
We wish to emphasize that our focus in this paper on a �method of
interpretation�which is bound by a �precedent system� (as well as to the
written law or contract, in case there are no contradictions or lacunae) is not
because we believe that it is more realistic or important than other methods,
but because unlike the others we have examined, it generates contradictions
in equilibrium. See Shavell (2006) and Posner (2005) for further discussion
of various constraints on the interpretation of contracts.
4.1 Maximal discretion
Let us relax the constraint that I is a measurable function of P c. Instead,
suppose that the Interpreter�s strategy is a function I : [0; 1) ! R that
assigns actions to states. We impose one constraint on this function: if
ack = a for every eck 2 Ec for which x 2 eck, then I(x) = a. That is, the
Interpreter cannot overrule an unambiguous instruction, but he has total
freedom in any other case. This variant of the model �ts situations in which
courts are not bound by precedents.
The rationale for contradictions disappears in this case. All that the
Writer needs to decide upon is the degree of vagueness �(c). Given the lack
of any constraint on the Interpreter�s discretion in case of vague instructions,
there is no need to distinguish between lacunae and contradictions: any state
over which the Interpreter has discretion will induce the outcome x+ a = m
with probability one. The Writer can attain the optimal degree of vagueness
with a lacuna, and without introducing any contradiction. Although the
equilibrium list can contain contradictions, there is no special reason for
them, and the list may contain only a lacuna.
The optimal degree of vagueness in this case is given by (1). When
the Interpreter has unlimited discretion in case of a lacuna, the Writer�s
20
expected utility conditional on delegation is precisely �m2. The reason is
that delegating an event to the Interpreter completely eliminates the variance
of x + a (conditional on this event). As we saw in Section 3.2, this leads to
expression (1), which is larger than the equilibrium degree of vagueness in
our original model.
Note that the outcome of this alternative model Pareto-dominates the
outcome given by the original model. This exposes a weakness of the legislature-
court interpretation. Recall that we interpret the assumption that I is mea-
surable with respect to P c as re�ecting a precedent system. But our analysis
in this sub-section reveals that such a system is Pareto-inferior to a sys-
tem without precedents. The question arises, what is the rationale for the
precedent system in the �rst place? In order to address this question, one
would have to enlarge the scope of the model and incorporate the costs of
interpreting contradictions and lacunae.
4.2 Minimal discretion
An alternative assumption, at the other extreme, is that when the Interpreter
faces contradictory instructions, he can only pick one of the actions speci�ed
by the instructions. Formally, suppose that I : [0; 1)! R is a function fromstates to actions, but assume that whenever J c(x) 6= ?, I(x) 2 fa 2 R j a =ack and x 2 eck for some kg.In this case, it is straightforward to see that there is no rationale for
contradictions. Suppose that x 2 eck; ecj. The Interpreter is forced to choosebetween ack and a
cj. Clearly, he will choose the action that is more favorable to
him, which may be less favorable for the Writer. Therefore, the instructions
list induces an e¤ective partition which prescribes no more than K di¤erent
actions (excluding the actions induced by a lacuna). If the Writer avoided
contradictions, he would be able to induce an e¤ective partition which pre-
scribes K di¤erent actions (excluding the actions induced by a lacuna), but
these actions are better for him than the ones chosen by the Interpreter.
21
5 Conclusion
Our objective in this paper was modest: to construct the simplest possible
model that is capable of accommodating contradictions in contracts, and to
provide a complexity-based rationale for contradictions. The basic idea is
that when writing complex contracts is costly, there may be an incentive to
introduce contradictions, and leave for another agent the task of interpreting
them. The e¤ect of complexity on the measure of delegated states is sen-
sitive to the complexity notion, as well as to the norms that constrain the
interpretation of contradictions. Hopefully, these ideas may inspire further
research on contract-theoretic models, in which the court actively interprets
laws and contracts.
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